Question

if $\int\frac{3x+4}{x^{3}-2x-4}dx =log\left|x-2\right| +k logf\left(x\right)+c$, then

• A. $f\left(x\right) =f\left(x\right) =\left|x^{2 }+2x +2\right|$
• B. $f\left(x\right) =x^{2}+2x+2$
• C. $k =-1/2$
• D. All of these

Answer 1

Answer: (D)

$\frac{3x+4}{x^{3}-2x -4} =\frac{ 3x +4}{\left(x-2\right)\left(x^{2}+2x+2\right)} $

$ =\frac{A}{x-2} +\frac{Bx+C}{x^{2}+2x+2} $

$\Rightarrow 3x +4 = A\left(x^{2}+2x+2\right) +\left(Bx+C\right)\left(x-2\right)$

$\therefore A + B = O', 2A-2B +C = 3, 2A-2C = 4,\Rightarrow A= 1, B= C=-1 $

$\therefore \int\frac{3x+4}{x^{3}-2x-4}dx =\int\frac{dx}{x-2}-\frac{1}{2} \int\frac{2x+2}{x^{2}+2x+2}dx$

$= log\left|x-2\right|-\frac{1}{2}log\left|x^{2}+2x+2\right|+C$

$\Rightarrow k=-\frac{1}{2} and f\left(x\right)=\left|x^{2}+2x+2\right| $